Consider the following statement For all real numbers r if r

Consider the following statement: For all real numbers r, if r^3 is irrational, than r is rational.

(a). Prove the statement using either proof by contradiction or by proof by contraposition. Clearly indicate which method you are using.

(b). If you used proof by contradiction in part (a), write what you would \"suppose\" and what you would \"show\" to prove the statement by contraposition. If you used proof by contraposition in part (a), write what you would \"suppose\" and what you would \"show\" to prove the statement by contradiction.

Solution

Let r = cube root(2), so r is irrational, but r^3 = 2, which is rational. Hence the statement is false.

By contradiction: Assume that r is rational. Then r = m/n for some integers m,n. And r^3 = m^3/n^3.
Since m^3 & n^3 are integers, this makes r^3 rational, which creates a contradiction, as r^3 is irrational...